POLARIZED LIGHT FROM DIFFERENT SOURCES. 243 /3, let the quantities sin2 (ft - /3), &c. be replaced by sines and cosines of multiple arcs, and let our equation be put under the form A + B cos 2/3 4- G sin 2/3 = 0. Then A, B, 0 must be separately equal to zero, or sin2 (ft - ft) cos2 (a, - a,) + cos2 (ft + ft) sin2 (a, -«,) = !;) cos 2ft cos 2 (a2 - a) + cos 2/3, cos 2 (eq - a) = 0; [ (14). sin 2ft + sin 2ft =0. j Replacing unity in the right-hand member of the first of these equations by cos'2 («2 - oq) + sin2 (a2 - cq), we find cos2 (ft - ft) cos2 (a, - cq) + sin2 (ft + ft) sin2 (a, - a,) = 0; whence ft = -ft, a^cq+900, or else ft and ft differ by 90°, and «2 = eq, except in the particular case in which ft =±45°, when ft = + 45° satisfies the equation independently of oc2. Hence the streams must be polarized oppositely, a condition which may always be expressed by £2=-ft> a2=ai+90°, which equations satisfy the second and third of equations (14) independently of a, as it might have been foreseen that they would, since it has been already shewn that the condition (12) is satisfied in the case of oppositely polarized streams. It now appears that it is only in the case of such streams that this is satisfied. 6. The properties of oppositely polarized pencils which have been proved, render it in a high degree probable that it is a general law that in a doubly refracting medium the two polarized pencils transmitted in a given direction are oppositely polarized. Were this not the case, the two pencils, polarized otherwise than oppositely, into which a polarized pencil is resolved on entering into the medium, would at emergence compound a pencil of which the intensity would depend upon the retardations of phase of one pencil relatively to the other, so that such a medium, when examined with polarized light, ought to exhibit rings or colours without the employment of an analyzer. It is here supposed 16-2